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Theorem elxp6 8009
Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 7913. (Contributed by NM, 9-Oct-2004.)
Assertion
Ref Expression
elxp6 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))

Proof of Theorem elxp6
StepHypRef Expression
1 elxp4 7913 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩ ∧ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))
2 1stval 7977 . . . . 5 (1st𝐴) = dom {𝐴}
3 2ndval 7978 . . . . 5 (2nd𝐴) = ran {𝐴}
42, 3opeq12i 4879 . . . 4 ⟨(1st𝐴), (2nd𝐴)⟩ = ⟨ dom {𝐴}, ran {𝐴}⟩
54eqeq2i 2746 . . 3 (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ↔ 𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩)
62eleq1i 2825 . . . 4 ((1st𝐴) ∈ 𝐵 dom {𝐴} ∈ 𝐵)
73eleq1i 2825 . . . 4 ((2nd𝐴) ∈ 𝐶 ran {𝐴} ∈ 𝐶)
86, 7anbi12i 628 . . 3 (((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶) ↔ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶))
95, 8anbi12i 628 . 2 ((𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) ↔ (𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩ ∧ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))
101, 9bitr4i 278 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107  {csn 4629  cop 4635   cuni 4909   × cxp 5675  dom cdm 5677  ran crn 5678  cfv 6544  1st c1st 7973  2nd c2nd 7974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-1st 7975  df-2nd 7976
This theorem is referenced by:  elxp7  8010  eqopi  8011  1st2nd2  8014  eldju2ndl  9919  eldju2ndr  9920  r0weon  10007  qredeu  16595  qnumdencl  16675  setsstruct2  17107  tx1cn  23113  tx2cn  23114  txhaus  23151  psmetxrge0  23819  xppreima  31871  ofpreima2  31891  smatrcl  32776  1stmbfm  33259  2ndmbfm  33260  oddpwdcv  33354  prproropf1olem0  46170
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